Approximation – JSS1 Mathematics Lesson Note (Second Term)

Approximation is the process of using rounded numbers to estimate the outcome of calculations. It helps us decide whether an answer to a calculation is right. Numbers can be approximated to decimal places, significant figures, or the nearest whole number, ten, hundred and so on.

Behavioural objectives

By the end of the lesson, students should be able to:

• state the number of decimal places in a decimal number;
• round a number to a given number of decimal places;
• identify the significant figures in a number;
• round a number to a given number of significant figures;
• round a decimal to the nearest tenth, hundredth, thousandth and whole number.

What are decimal places?

A number such as 197.7658 is a decimal number. The whole number part is 197, and the decimal part is 7658. The point between them is called the decimal point.

To find the number of decimal places (d.p.), count the figures after the decimal point. The number 197.7658 therefore has 4 d.p.

How do we round a number to decimal places?

To correct a number to a given number of decimal places:

1. Look for the required decimal place. Start counting from the figure after the point.
2. Look at the next figure to the right (the decider).
3. If the decider is 5 or more, round up. If it is less than 5, add nothing.

Example 1

Give the number 78.05624 correct to (a) 1 d.p., (b) 2 d.p. and (c) 3 d.p.

Solution:

• (a) 78.1. The 1st decimal place is 0 (zero counts as a figure here). The decider is 5, so round the 0 up to 1.
• (b) 78.06. The 2nd decimal place is 5. The decider is 6, so round the 5 up to 6.
• (c) 78.056. The 3rd decimal place is 6. The decider is 2, which is less than 5, so add nothing.

Example 2

Give 57.9945 correct to (a) 2 d.p. and (b) 4 d.p.

Solution:

• (a) 57.99 (2 d.p.). The decider is 4, so add nothing.
• (b) 57.9945 (4 d.p.). The number already has 4 d.p., so it stays the same.

What are significant figures?

The word “significant” means “important”. We write significant figures as s.f. The first significant figure is always the first non-zero figure when you read a number from the left.

Numbers greater than one

The number 865 034 has six figures. The first figure from the left, 8, is worth 800 000 (place value), so it is the most significant figure. It is therefore the 1st significant figure, and 4 is the least significant, or 6th, significant figure.

Numbers less than one

The number 0.000007685 is given to 9 decimal places. The zero before the decimal point means there are no units. The 5 zeros after the decimal point are not significant figures.

The first significant figure is therefore 7, followed by 6 (2nd s.f.), 8 (3rd s.f.) and 5 (4th s.f.).

How do we round a number to significant figures?

To correct a number to a given number of significant figures:

1. Look for the required significant figure.
2. Look at the next figure to the right (the decider).
3. If the decider is 5 or more, round up. If it is less than 5, add nothing.

Replace the figures after the required significant figure with zeros if they come before the decimal point.

Example 3

Give 4540 correct to (a) 1 s.f., (b) 2 s.f. and (c) 3 s.f.

Solution:

• (a) 4540 = 5000. The 1st s.f. is 4. The decider is 5, so round the 4 up to 5. The other figures become zeros.
• (b) 4540 = 4500. The 2nd s.f. is 5. The decider is 4, which is less than 5, so add nothing. The remaining figures become zeros.
• (c) 4540 = 4540. The 3rd s.f. is 4. The decider is 0, so add nothing.

Example 4

Convert 0.00005791 to (a) 2 s.f. and (b) 3 s.f.

Solution:

• (a) 0.000058. The first two significant figures are 5 and 7. The decider is 9, so round the 7 up to 8.
• (b) 0.0000579. The first three significant figures are 5, 7 and 9. The decider is 1, so add nothing.

How do we round decimals to the nearest tenth, hundredth, thousandth and whole number?

The tenth is the 1st decimal place, the hundredth is the 2nd and the thousandth is the 3rd. Counting starts after the decimal point.

Example 5

Give 474.4547 correct to the nearest (a) tenth, (b) hundredth and (c) thousandth.

Solution:

• (a) 474.4547 = 474.5. The decider is 5, so round the 4 up to 5.
• (b) 474.4547 = 474.45. The decider is 4, so add nothing.
• (c) 474.4547 = 474.455. The decider is 7, so round the 4 up to 5.

Example 6

Round off (a) 13.73 and (b) 34.245 to the nearest whole number.

Solution:

• (a) 13.73 = 14. The whole number part is 13. The first decimal figure is 7, which is 5 or more, so round 13 up to 14.
• (b) 34.245 = 34. The first decimal figure is 2, which is less than 5, so the whole number stays 34.

Evaluation

1. How many decimal places has 197.7658?
2. Give 78.05624 correct to 4 d.p.
3. Give 0.00005791 correct to 1 s.f.
4. Give 474.4547 correct to the nearest whole number.
5. Give 34.245 correct to the nearest tenth.

Assignment

1. Give 57.9945 correct to 1 d.p.
2. Give 865 034 correct to 3 s.f.
3. Give 0.000007685 correct to 2 s.f.
4. Round 13.73 to the nearest tenth.